Large-scale tomographic particle image velocimetry using helium-filled soap bubbles

Abstract

To measure large-scale flow structures in air, a tomographic particle image velocimetry (tomographic PIV) system for measurement volumes of the order of one cubic metre is developed, which employs helium-filled soap bubbles (HFSBs) as tracer particles. The technique has several specific characteristics compared to most conventional tomographic PIV systems, which are usually applied to small measurement volumes. One of them is spot lights on the HFSB tracers, which slightly change their position, when the direction of observation is altered. Further issues are the large particle to voxel ratio and the short focal length of the used camera lenses, which result in a noticeable variation of the magnification factor in volume depth direction. Taking the specific characteristics of the HFSBs into account, the feasibility of our large-scale tomographic PIV system is demonstrated by showing that the calibration errors can be reduced down to 0.1 pixels as required. Further, an accurate and fast implementation of the multiplicative algebraic reconstruction technique, which calculates the weighting coefficients when needed instead of storing them, is discussed. The tomographic PIV system is applied to measure forced convection in a convection cell at a Reynolds number of 530 based on the inlet channel height and the mean inlet velocity. The size of the measurement volume and the interrogation volumes amount to 750 mm × 450 mm × 165 mm and 48 mm × 48 mm × 24 mm, respectively. Validation of the tomographic PIV technique employing HFSBs is further provided by comparing profiles of the mean velocity and of the root mean square velocity fluctuations to respective planar PIV data.

Appendix

The line integral through a spherically symmetric interpolation kernel f(r) is equivalent to the Abel transform of the function (Bracewell 1978). The Abel transform of f(r) is defined as

$$ f_{A} (d) = \int\limits_{d}^{\infty } {{\frac{f(r)r{\text d}r}{{\sqrt {r^{2} - d^{2} } }}}} $$

(7)

where r is the radius and d the minimal distance between the centre of the function and the integration path (see Fig. 6).

In the present investigation, a bilinear interpolation kernel

$$ f(r) = \left\{ {\begin{array}{*{20}c} {1 - r/r_{\max } } \\ 0 \\ \end{array} } \right.\begin{array}{*{20}c} {r \le r_{\max } } \\ {\text{else}} \\ \end{array} $$

(8)

is used. Therby r _max is the radius of the filter and chosen to be equal to the side length of a voxel. The Abel transform of the bilinear interpolation kernel is finally

$$ f_{A} (d) = \sqrt {r_{\max }^{2} - d^{2} } - {\frac{{d^{2} }}{{r_{\max } }}} \cdot \ln \left| {r_{\max } + \sqrt {r_{\max }^{2} - d^{2} } } \right| + {\frac{{d^{2} }}{{r_{\max } }}} \cdot \ln \left| d \right| $$

(9)

whereas the upper integration limit is set to r _max. In order to have a line integral, respectively, weighting coefficient between 0 and 1 the value is normalised with f _A (d = 0). Using this transformation, it is assumed that the integration is performed along a straight line.